No luck with the search engine: a recruiter tells me not to worry that I will work on an L-visa for a possible 3-,6-, or 9-month contract coz all their work is in government schools. Is that kosher? Thanks in advance for the information.

Do NOT come on ANYTHING other than a legit visa.
Government schools must still have the authority to employ foreigners.
No contract will be enforceable if you are working on a lie ie a tourist visa.
Even legit contracts are almost impossible to enforce.
My advice to anyone on their first job is look for a least downside and you're lining yourself to be burned with the approach you describe.

Yes, I see. Thanks everyone so far. I kind of sensed something was amiss when I asked a few reasonable questions like if I needed to get a medical exam, or visit the Chinese Embassy for an L visa, and the only response I got was a hearty thanks for my continued interest and a request to send my certificate scans. I replied, "But you haven't answered any of my questions," and now they have fallen silent for two days. Hmm...

This might be a location problem, but you can get jobs with a Z visa. In fact, some schools are only aware of the invitation letter process. They don't realize you can transfer to a new school without needing all the new paperwork.

Also, another red flag is that when a recruiter tells you one rule for all of China it is most likely not true. Each area has their own requirements.